Abstract
The symmetric group \(S_n\) on \(n\) letters is a metric space with respect to the Hamming distance. The corresponding isometry group is well known to be isomorphic to the wreath product \(S_n \wr S_2\). A subset of \(S_n\) is called a permutation code or a permutation array, and the largest possible size of a permutation code with minimum Hamming distance \(d\) is denoted by \(M(n, d)\). Using exhaustive search by computer on sets of orbits of isometry subgroups \(U\) we are able to determine serveral new lower bounds for \(M(n,d)\) for \(n \le 22\). The codes are given by the group \(U\) and representatives of the \(U\)-orbits.
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The third author was supported in part by the Academy of Finland under Grant No. 132122.
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Communicated by K. Metsch.
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Janiszczak, I., Lempken, W., Östergård, P.R.J. et al. Permutation codes invariant under isometries. Des. Codes Cryptogr. 75, 497–507 (2015). https://doi.org/10.1007/s10623-014-9930-z
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DOI: https://doi.org/10.1007/s10623-014-9930-z